BRASÍLIA—Brazil’s top prosecutor filed criminal charges against President Michel Temer on Monday, marking a critical new phase for the corruption crackdown that has roiled Latin America’s largest country for the past three years.

Attorney General Rodrigo Janot brought charges of corruption against Mr. Temer, alleging the leader took about $150,000 in bribes from the former chairman of meatpacking giant JBS SA, the world’s biggest...